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A deterministic compartmental model is developed for examining the dynamics of digesta marker outflows in animals, focussing on models which will simulate bimodal kinetics with two peaks in the time course of marker outflow. First, to establish the background to the subsequent modelling of bimodal flow, we examine the compartmental scheme which gives rise mechanistically to the gamma function, and derive or describe various useful properties. This is illustrated by varying two key parameters of the gamma function, n (the number of compartments), and k (the rate constant for compartment emptying). Next, a more articulated compartmental scheme is constructed, and by progressive parameter changes, it is shown how bimodal faecal marker outflow can be achieved. Last, progressive simplification is applied to this scheme to arrive at what is (hopefully) the simplest compartmental scheme which can be used to simulate bimodal kinetics. This may be used mechanistically to describe the role of digesta flow in animals which exhibit such characteristics., (Copyright © 2017 Elsevier Ltd. All rights reserved.)

A deterministic mathematical model is developed for the dynamics of bluetongue disease within a single farm. The purpose is to examine widely the possible behaviours which may occur. This is important because of the increasing impact of blue tongue due to global warming. The model incorporates a recently suggested modification of logistic growth for the vectors which can greatly affect early disease dynamics and employs a variable number of up to 10 sequential pools for incubating vectors and for incubating and infectious hosts. Ten sequential pools represent the possible loss of immunity of recovered hosts over a 3-year period. After formally describing the model, the impact of the two logistic growth scenarios considered is examined in Section 3.1. The scenarios are applied with parameters that give identical long-term consequences but the early dynamics can be greatly affected. In the two scenarios, the effect of varying the assumed constant birth rate (scenario 1) or constant mortality rates (scenario 2) is considered. If the recovered (and immune) hosts, are assumed to lose their immunity, then, given particular values of the host-vector coupling constants, the system can exhibit autonomous oscillations (Section 3.2). Seasonality is represented by air temperature, and it is assumed that air temperatures below a threshold can increase vector mortality (Section 3.3). Adding seasonal effects on mortality to the autonomous oscillations resulting from recovered and resistant hosts losing immunity can give rise to chaos (Section 3.4). This could help explain the unusual persistence and re-occurrence of the disease. Finally (Section 3.5), the roles of host birth and mortality rates in examined, particularly in relation to placental transmission of the virus to offspring. It is concluded that the latter does not make an appreciable contribution to disease dynamics., (Copyright © 2016 Elsevier Ltd. All rights reserved.)

Treating resource allocation within plants, and between plants and associated organisms, is essential for plant, crop and ecosystem modelling. However, it is still an unresolved issue. It is also important to consider quantitatively when it is efficient and to what extent a plant can invest profitably in a mycorrhizal association. A teleonomic model is used to address these issues. A six state-variable model giving exponential growth is constructed. This represents carbon (C), nitrogen (N) and phosphorus (P) substrates with structure in shoot, root and mycorrhiza. The shoot is responsible for uptake of substrate C, the root for substrates N and P, and the mycorrhiza also for substrates N and P. A teleonomic goal, maximizing proportional growth rate, is solved analytically for the allocation fractions. Expressions allocating new dry matter to shoot, root and mycorrhiza are derived which maximize growth rate. These demonstrate several key intuitive phenomena concerning resource sharing between plant components and associated mycorrhizae. For instance, if root uptake rate for phosphorus is equal to that achievable by mycorrhiza and without detriment to root uptake rate for nitrogen, then this gives a faster growing mycorrhizal-free plant. However, if root phosphorus uptake is below that achievable by mycorrhiza, then a mycorrhizal association may be a preferred strategy. The approach offers a methodology for introducing resource sharing between species into ecosystem models. Applying teleonomy may provide a valuable short-term means of modelling allocation, avoiding the circularity of empirical models, and circumventing the complexities and uncertainties inherent in mechanistic approaches. However it is subjective and brings certain irreducible difficulties with it., (Crown Copyright © 2013 Published by Elsevier Ltd All rights reserved.)

A simple generic model of foot and mouth disease (FMD) is presented. The dynamics of FMD at the index site (the farm where the initial infection occurs) is represented. Spread of disease from the index site is simulated with a three-term radial function, the terms corresponding to natural processes, short- and long-range movements. Parameterization includes the processes of threshold for clinical disease reporting, time delay before action is initiated, removal (slaughter) rate at the index farm; movement restrictions, culling of farms out to a given radius (from the index site), rate of culling, and vaccination (timing; rate of build-up of immunity). Between-farm basic reproductive ratio, R0bf, the number of primary (direct) infected farms arising from the initially infected (index) farm during the lifetime of the infection at the index farm, is calculated numerically. A "predictor-corrector" method is developed to estimate an upper limit to the number of secondary infections which could arise during this time. The model is easily programmed and is quick to execute. Simulation results are discussed in relation to the processes assumed in the model. They suggest that, given current UK farm practice, early disease detection, combined with immediate rapid slaughtering at the index site and restriction of short-range and long-range movements are relatively effective strategies of disease control. However, farm practice changes continually, and any control strategies need re-assessment in the light of current farm practice when and where an FMD outbreak occurs. A transparent and "simple" model can facilitate this process. The uncorrected model, without tuning, is compared with the cattle data from the 2001 UK FMD epidemic (Section 6; Fig. 8).